We experimentally demonstrate the enhancement of the spontaneous emission rate of GaAs quantum wells embedded in rolled-up metamaterials. We fabricate microtubes whose walls consist of alternating Ag and (In)(Al)GaAs layers with incorporated active GaAs quantum-well structures. By variation of the layer thickness ratio of the Ag and (In)(Al)GaAs layers we control the effective permittivity tensor of the metamaterial according to an effective medium approach. Thereby, we can design samples with elliptic or hyperbolic dispersion. Time-resolved low temperature photoluminescence spectroscopy supported by finite-difference time-domain simulations reveal a decrease of the quantum well's spontaneous emission lifetime in our metamaterials as a signature of the crossover from elliptic to hyperbolic dispersion.